Folding an n × n checkerboard pattern from a
square of paper that is white on one side and black on the other has been
thought for several years to require a paper square of
semiperimeter n2. Indeed, within a restricted class of
foldings that match all previous origami models of this flavor, one can prove
a lower bound of n2 (though a matching upper bound was
not known). We show how to break through this barrier and fold an
n × n checkerboard from a paper square of
semiperimeter ½ n2 + O(n).
In particular, our construction strictly beats semiperimeter
n2 for (even) n > 16, and for
n = 8, we improve on the best seamless folding.